Questions: Find the equation of the tangent line to the curve y=(3 ln (x)) / x at the points at the point (1,0) y= at the point (e, 3 / e) y=

Solution Steps

To find the equation of the tangent line to the curve \( y = \frac{3 \ln(x)}{x} \) at given points, we need to:

1. Compute the derivative of the function \( y \) to get the slope of the tangent line.
2. Evaluate the derivative at the given points to find the slope at those points.
3. Use the point-slope form of the equation of a line to write the equation of the tangent line at each point.

Given the function \( y = \frac{3 \ln(x)}{x} \), we first compute its derivative to find the slope of the tangent line. The derivative is: \[ y' = \frac{d}{dx} \left( \frac{3 \ln(x)}{x} \right) = -\frac{3 \ln(x)}{x^2} + \frac{3}{x^2} \]

We need to find the slope of the tangent line at the points \((1, 0)\) and \((e, \frac{3}{e})\).

1. At the point \((1, 0)\): \[ y'(1) = -\frac{3 \ln(1)}{1^2} + \frac{3}{1^2} = 0 + 3 = 3 \]

2. At the point \((e, \frac{3}{e})\): \[ y'(e) = -\frac{3 \ln(e)}{e^2} + \frac{3}{e^2} = -\frac{3 \cdot 1}{e^2} + \frac{3}{e^2} = 0 \]

Using the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), we can write the equations of the tangent lines.

1. At the point \((1, 0)\) with slope \( m = 3 \): \[ y - 0 = 3(x - 1) \] Simplifying, we get: \[ y = 3x - 3 \]

2. At the point \((e, \frac{3}{e})\) with slope \( m = 0 \): \[ y - \frac{3}{e} = 0(x - e) \] Simplifying, we get: \[ y = \frac{3}{e} \]

Final Answer